Week 11 0B2 Tutorial

• An explanation of when solutions are increasing and decreasing.

• A sketch of solutions of the ODE.

• A sketch of the tangents of solutions of the ODE.

• A sketch of F(x,y).

• When F(x,y) is not defined at (x,y)=(a,b).

• When F(a,b)=0.

What is the point in a direction field?

• We can obtain rough sketch of any given solution without solving the ODE.

• It helps us to see what solutions "look like" even if we do not know how to solve the ODE.

• Can ascertain properties/qualities of the solutions (without solving the ODE).

• It gives us a solution to the ODE.

• A graph sketch of the solution y(x) such that y(a)=b that follows the direction field.

• A solution of the ODE.

• The inclination of solutions of the ODE.

• Graph sketches of solutions of the ODE.

• Functions y(x) satisfying the equation F(x,y)=c for some constant c.

• The graph of the equation F(x,y)=c for some constant c.

Why can isoclines be useful?

• Can be used to extend the direction to problematic points discussed in question

• We can use the isocline to derive a solution of the ODE.

• Isoclines are solutions to the ODE.

• We can use them to obtain a more accurate direction field.